Electromagnetic NDE Lecture Slides

1

Electromagnetic NDE

Peter B. Nagy

Research Centre for NDE

Imperial College London

2011

Aims and Goals

Aims

1 The main aim of this course is to familiarize the students with Electromagnetic (EM) Nondestructive Evaluation (NDE) and to integrate the obtained specialized knowledge into their broader understanding of NDE principles.

2 To enable the students to judge the applicability, advantages, disadvantages, and technical limitations of EM techniques when faced with NDE challenges.

Objectives

At the end of the course, students should be able to understand the:1 fundamental physical principles of EM NDE methods2 operation of basic EM NDE techniques3 functions of simple EM NDE instruments4 main applications of EM NDE

2

Syllabus

1 Fundamentals of electromagnetism. Maxwell's equations. Electromagnetic wave propagation in dielectrics and conductors. Eddy current and skin effect.

2 Electric circuit theory. Impedance measurements, bridge techniques. Impedance diagrams. Test coil impedance functions. Field distributions.

3 Eddy current NDE techniques. Instrumentation. Applications; conductivity, permeability, and thickness measurement, flaw detection.

4 Magnetic measurements. Materials characterization, permeability, remanence, coercivity, Barkhausen noise. Flaw detection, flux leakage testing.

5 Alternating current field measurement. Alternating and direct current potential drop techniques.

6 Microwave techniques. Dielectric measurements. Thermoelectric measurements.

7 Electromagnetic generation and detection of ultrasonic waves, electromagnetic acoustic transducers (EMATs).

3

1 Electromagnetism

1.1 Fundamentals

1.2 Electric Circuits

1.3 Maxwell's Equations

1.4 Electromagnetic Wave Propagation

1.1 Fundamentals of Electromagnetism

4

Electrostatic Force, Coulomb's Law

x

z

y

r

Q2

Q1

Fe

Fe

Fe Coulomb force

Q1, Q2 electric charges (± ne, e ≈ 1.602 × 10-19 As)

er unit vector directed from the source to the target

r distance between the charges

ε permittivity (ε0 ≈ 8.85 × 10-12 As/Vm)

1 2e 24 x

Q dQ xdrr

=πε

F e

2 , 2dQ q dA dA d= = πρ ρ

1e 3

02xQ q x d

r

∞

ρ=

ρ ρ= ∫

εeF

2 22 2

, d r rr xdr r x

ρρ = − = =

ρ−

1e 22

x

r x

Q q x drr

∞

== ∫

εeF

1e 2

xQ q=

εeF

x

dQ2

Q1

Fe

ρ

dρ

r

infinite wall of uniform charge

density q

independent of x

1 2e 24 r

Q Qr

=πε

F e

Electric Field, Plane Electrodes

Qt

Fe

x

z

y

e 2t xQ q

=εeF

infinite wall of uniform charge density q

2 xq

=ε

E e

E+Q -Q

A

QqA

=

charged parallel plane electrodes Q

xq

≈ε

E e

e tQ=F E

5

e tQ=F E

Electric Field, Point Sources

e 24s t

rQ Q

r=

πεF e

24s

rQ

r=

πεE e

monopole

+Qs

+Qs

-Qs

1 32sQ dE

r≈

πε

2 34sQ dE

r≈

πε

+Qs

-Qs

d

E1

E2

E1

dipole

Electric Field of Dipole

z z R RE E= +E e e

3/ 2 3/ 22 2 2 2

/ 2 / 24 ( / 2) ( / 2)

sz

Q z d z dEz d R z d R

⎡ ⎤− +⎢ ⎥= −⎢ ⎥πε ⎡ ⎤ ⎡ ⎤− + − +⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

( )23 3cos 1

4s

zQ dE

r≈ θ −

πε

3/ 2 3/ 22 2 2 24 ( / 2) ( / 2)s

RQ R RE

z d R z d R

⎡ ⎤⎢ ⎥= −⎢ ⎥πε ⎡ ⎤ ⎡ ⎤− + − +⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

33 sin 28

sR

Q dEr

≈ θπε

2 2 2r z R= + cosz r= θ sinR r= θ

R

z

+Qs

-Qs

d

θ

rr+

r

PEz

ER

E

6

Electric Dipole in an Electric Field

+Q

-Q

pe

Fe

E EE=E e

e dQ Q d= =p d e

e e Q= × = ×T d F d E

e Q=F E

pe electric dipole moment

Q electric charge

d distance vector

E electric field

Fe Coulomb force

Te twisting moment or torque

Fe e e= ×T p E

Electric Flux and Gauss’ Law

q charge (volume) density

D electric flux density (displacement)

E electric field (strength, intensity)

ε permittivity

ψ electric flux

Qenc enclosed charge

closed surface S

D

dS

Qenc

Sdψ = ∫∫ D Si

= εD E

d dψ = D Si

Sd dS=S e

encV

q dV Qψ = =∫∫∫

q∇ =Di

7

Electric Potential

W work done by moving the charge

Fe Coulomb force

ℓ path length

E electric field

Q charge

U electric potential energy of the charge

V potential of the electric field

E

QFe

dℓ

A

BB A ABU U U WΔ = − =

edW = − F i dℓ

BAB

AW Q= − ∫ Eidℓ

U V Q=

BB A

AV V VΔ = − = − ∫ Eidℓ

Capacitance

+Q

-Q

E

C capacitance

V voltage difference

Q stored charge

Q CV=

+

-

S+ -

SV V V= − = − ∫ Eidℓ

QCV

=

E

+Q

-Q

A

-Q

E

+Q

dℓ

QDA

ACDE

V E

⎫≈ ⎪⎪ ε⎪ ≈⎬= ⎪ε⎪⎪≈ ⎭

8

Current, Current Density, and Conductivity

I currentQ transferred charget timeJ current densityA cross section arean number density of free electronsvd mean drift velocitye charge of protonm mass of electronτ collision timeΛ free pathv thermal velocityk Boltzmann’s constantT absolute temperatureσ conductivity

dQIdt

=

dI d= J Ai

I d= ∫∫ J Ai

dne= −J v

ddQ ne d dt= − v Ai

dm e= −τv E

vΛ

τ =

21 32 2

mv kT=

E

dA

2nem

τ= = σJ E E

Resistivity, Resistance, and Ohm’s Law

V voltage

I current

R resistance

P power

σ conductivity

ρ resistivity

L length

A cross section area

I

+_V

A

dℓ

0 0

L Ld dRA A

ρ= =∫ ∫

σ

i i

i

LRA

ρ= ∑

1ρ =

σ

LRA

ρ=

+

-

S+ -

SV V V= − = − ∫ Eidℓ

0 0

L LJ dV d I I RA

= = =∫ ∫σ σ

VRI

=

dU dQP V V Idt dt

= = =

9

Magnetic Field

BQ

Fm

dv

e Q=F E

m Q= ×F v B

( )Q= + ×F E v B

F Lorenz forcev velocityB magnetic flux densityQ charge

+I -I

B

pm magnetic dipole moment(no magnetic monopole)

N number of turnsI currentA encircled vector area

m N I=p A

pm

Magnetic Dipole in a Magnetic Field

m Q= ×F v B

pm magnetic dipole moment

Q charge

v velocity

R radius vector

B magnetic flux density

Fm magnetic force

Tm twisting moment or torque

m N I=p A

+I

-Ipm

Fm

B

Fm

2m 2 r v

Qv RR

= π ×π

p e e

2A R= πQN I =τ

2 Rvπ

τ =

m12

Q= ×p R v

m m12

= ×T R F

22

m m m0

1 1cos2 2

T R F d R Fπ

= α α =∫π

m m= ×T p B

10

Magnetic Field Due to Currents

2 34 4s s

rQ Q

r r= = =

πε πεE e rCoulomb Law:

= εD E

= μB H

Biot-Savart Law: 2 34 4rI d Id

r r= × = ×

π πH e e rdℓ

dℓ

I

dℓ r

HH magnetic field

μ magnetic permeability

24 rI d

r= ×∫

πH e e

Ampère’s Law

24 rI dd

r= ×

πH e e

encS

d Q=∫∫ D SiGauss’ Law:

infinite straight wire

2 2 2 3/ 244 ( )I d R I R dd

rr Rθ θ= =ππ +

H e e

2 2 3/ 202 2( )

I R d IHRR

∞θ = =∫

π π+

2H ds H R Iθθ = π =∫

dℓ

I

dℓ

R

Hrℓ

s

2IH

Rθ =π

Biot-Savart Law:

Ampère’s Law:

Ampère’s Law: encd I=∫ H si

∇× =H J

11

N IΦ = μ Λ

Є dV NdtΦ

= − =

Induction, Faraday’s Law, Inductance

E induced electric field


t time

Є induced electromotive force

s boundary element of the loop

Φ magnetic flux

S surface area of the loop

I N

Φ

V

μ magnetic permeability

N number of turns

I current

Λ geometrical constant

L (self-) inductanceI LN

Φ =

2L N= μ Λ

SdΦ = ∫∫B Si

Є ddtΦ

= −

t∂

∇× = −∂BE

dIV Ldt

=

B

Є d= ∫ E si

ЄS

dt

∂= − ∫∫

∂B Si

Electric Boundary ConditionsFaraday's law:

t∂

∇× = −∂BE

Gauss' law:

q∇ =Di

xt

medium I

medium II

DIθΙ

boundary

DII

DII,t

DII,n

θΙΙ

DI,n

DI,t

xn

xt

medium I

medium II

EI

θΙ

EIIEI,t EII,n

θΙΙ

EI,n

EII,t

xn

I,n II,nD D=

I I,n II II,nE Eε = ε

I,t II,tE E=

I I,n II II,ntan tanE Eθ = θ

I III II

tan tanθ θ=

ε ε

tangential component of the electric field E is continuousnormal component of the electric flux density D is continuous

12

Magnetic Boundary ConditionsAmpère's law:

t∂

∇× = +∂DH J

Gauss' law:

0∇ =Bi

xt

medium I

medium II

BIθΙ

boundary

BII

BII,t

BII,n

θΙΙ

BI,n

BI,t

xn

xt

medium I

medium II

HI

θΙ

HII

HII,t

HII,n

θΙΙ

HI,n

HI,t

xn

I,n II,nB B=

I I,n II II,nH Hμ = μ

I,t II,tH H=

I I,n II II,ntan tanH Hθ = θI II

I II

tan tanθ θ=

μ μ

tangential component of the magnetic field H is continuousnormal component of the magnetic flux density B is continuous

1.2 Electric Circuits

13

Є

Electric Circuits, Kirchhoff’s Laws

Є electromotive forceVi potential drop on ith element

Kirchhoff’s junction rule (current law):

Kirchhoff’s loop rule (voltage law):

0iV =∑

0=∫ EidℓI

+_

1R 2R

4R

3R1V 2V

4V3V0V

0iI =∑

encS

Q d= ∫∫ D Si

Ii current flowing into a junction from the ith branch

+_Є

1I 2I

4I

1R 2R

4R

3R

Circuit Analysis

Loop Currents:

Kirchhoff’s Laws:

+_Є

1I 2I

4I

1R 2R

3R1V 2V

3V0V

+_Є

1I 2I

4I

1R 2R

4R

3R1i 2i

4R

4V1 2 41 2 4

0V V VR R R

− − =

1 4 0 0V V V+ − =

2 3 4 0V V V+ − =

322 3

0VVR R

− =

1 1 1 2 4 0( ) 0i R i i R V+ − − =

2 2 2 3 1 2 4( ) 0i R i R i i R+ − − =

14

DC Impedance Matching

2g

2g g, where

(1 )

V RPR R

ξ= ξ =

+ ξ

22 V

P I V I RR

= = =

g g

g gand

V V RI V

R R R R= =

+ +

2g

3g

1(1 )

VdPd R

− ξ=

ξ + ξ

2g

max gg

when4V

P R RR

= =

_ VgV

gR

R+

P IV=W QV=

AC ImpedanceI

V dIV Ldt

=

I

V 1V I dtC

= ∫

I

V V R I=

VZ i LI

= = ωVZ RI

= =1VZ

I i C= =

ω

0

0ZiVZ R i X Z e

Iϕ= = + =

0 2 20

VZ R XI

= = +

-1arg( ) - tanZ V IXZR

= ϕ = ϕ ϕ =

( )0 0( ) Ii t i tI t I e I eω + ϕ ω= =

( )0 0( ) Vi t i tV t V e V eω + ϕ ω= =

ReI I=

ReV V=

0 0 IiI I e ϕ=

0 0 ViV V e ϕ=

15

AC Power

ReI I=( )0 0( ) Ii t i tI t I e I eω + ϕ ω= =0( ) cos( )II t I t= ω + ϕ

ReV V=( )0 0( ) Vi t i tV t V e V eω + ϕ ω= =0( ) cos( )VV t V t= ω + ϕ

* *0 01 1( ) ( )2 2

P I t V t I V= =( ) ( )P I t V t= ReP P=

( )0 0

12

I ViP I V e ϕ − ϕ=0 01 cos( )2 I VP I V= ϕ − ϕ

real notation complex notation correspondence

cos( ) cos cos sin sinα + β = α β − α β

cos( ) cos cos sin sinα − β = α β + α β

1 1cos( ) cos( ) cos cos2 2

α + β + α − β = α β

cos sinie iα = α + α

reminder:

AC Impedance Matching

VgV

gZ

Z≈

ReP P=

2 *g*

*g g

1 Re Re2 2 ( )( )

V ZP I VZ Z Z Z

⎧ ⎫⎪ ⎪= = ⎨ ⎬+ +⎪ ⎪⎩ ⎭

( )*g g g,Z Z R R X X= = = −

2gmax

g8V

PR

=

2g

2Re2 4

g g

g

V R i XP

R

⎧ ⎫−⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

16

1.3 Maxwell's Equations

Vector Operations

( )0

limSS S→

⎧ ⎫∫∇× = ⎨ ⎬⎩ ⎭

AA e ii dℓCurl of a vector:

0lim yS x z

V

dAA A

V x y z→

⎧ ⎫∫∫ ∂∂ ∂⎪ ⎪∇ = = + +⎨ ⎬ ∂ ∂ ∂⎪ ⎪⎩ ⎭

A SA

iiDivergence of a vector:

x y zx y z∂φ ∂φ ∂φ

∇φ = + +∂ ∂ ∂

e e eGradient of a scalar:

2 2 22

2 2 2x y z∂ φ ∂ φ ∂ φ

∇ φ = + +∂ ∂ ∂

Laplacian of a scalar:

2 2 2 2x x y y z zA A A∇ = ∇ + ∇ + ∇A e e eLaplacian of a vector:

2( ) ( )∇× ∇× = ∇ ∇ ⋅ − ∇A A AVector identity:

x y zx y z∂ ∂ ∂

∇ = + +∂ ∂ ∂

e e eNabla operator:

2 2 22

2 2 2x y z∂ ∂ ∂

∇ = ∇ ∇ = + +∂ ∂ ∂

iLaplacian operator:

y yx xz zx y z

A AA AA Ay z z x x y

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂⎛ ⎞∇× = − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠A e e ea

17

Maxwell's Equations

Ampère's law:

Faraday's law:

Gauss' law:

Gauss' law:

t∂

∇× = +∂DH J

t∂

∇× = −∂BE

q∇ =Di

0∇ =Bi

Field Equations:

conductivity = σJ E

permittivity = εD E

permeability = μB H

Constitutive Equations:

(ε0 ≈ 8.85 × 10-12 As/Vm)

(µ0 ≈ 4π × 10-7 Vs/Am)0 rμ = μ μ

0 rε = ε ε

1.4 Electromagnetic Wave Propagation

18

Electromagnetic Wave Equation

Maxwell's equations:

( )it

∂∇× = + = σ+ ωε

∂DH J E

it

∂∇× = − = − ωμ

∂BE H

0∇⋅ =E

0∇⋅ =H

( ) ( )i i∇× ∇× = − ωμ σ + ωεH H

( ) ( )i i∇× ∇× = − ωμ σ + ωεE E

2( ) ( )∇× ∇× = ∇ ∇ ⋅ − ∇A A A

2 ( )i i∇ = ωμ σ + ωεE E

2 ( )i i∇ = ωμ σ + ωεH H

2 ( )k i i= − ωμ σ + ωε

2 2( )k∇ + =E 0

2 2( )k∇ + =H 0

( )0

i t k xy y yE E e ω −= =E e e

( )0

i t k xz z zH H e ω −= =H e e

Example plane wave solution:

Wave equations:

Harmonic time-dependence: 0 0andi t i te eω ω= =E E H H

Wave Propagation versus Diffusion

Propagating wave in free space:

/ ( / )0

x i t xyE e e− δ ω − δ=E e

/ ( / )0

x i t xzH e e− δ − ω − δ=H e

Diffusive wave in conductors:

kcω

=

0 0

1c =μ ε

1 ik i= − ωμσ = −δ δ

1f

δ =π μσ

( / )0

i t x cyE e ω −=E e

( / )0

i t x czH e ω −=H e

2 ( )k i i= − ωμ σ + ωε

δ standard penetration depth

c wave speed

k wave number

Propagating wave in dielectrics:

d0 0 r

1c =μ ε ε r

d

cnc

= = ε

n refractive index

19

Intrinsic Wave Impedance( )

0i t k x

y y yE E e ω −= =E e e ( )0

i t k xz z zH H e ω −= =H e e

( )it

∂∇× = + = σ+ ωε

∂DH J E

( )0

z i t k xy y

H i k H ex

ω −∂∇ × = − =

∂H e e

( )k i i= − ωμ σ + ωε

Propagating wave in free space:0

00

377μη = ≈ Ω

ε

Propagating wave in dielectrics:0 0

0 r nμ η

η = ≈ε ε

Diffusive wave in conductors:1i iωμ +

η = =σ σδ

0

0

E iH i

ωμη = =

σ+ ωε

PolarizationPlane waves propagating in the x-direction:

( ) ( )0 0

i t k x i t k xy y z z y y z zE E E e E eω − ω −= + = +E e e e e

( ) ( )0 0

i t k x i t k xz z y y z z y yH H H e H eω − ω −= + = +H e e e e

0 00

0 0

y z

z y

E EH H

η = = −

0 0 0 0y zi iy y z zE E e E E eφ φ= =

y

z

y

z

y

z

Ey

EzE

0º (or 180º)y zφ − φ =

linear polarization elliptical polarization

90º (or 270º)y zφ − φ =

circular polarization

E E

20

Reflection at Normal Incidence

x

y

incident

reflected transmitted

I( )i i0

i t k xyE e ω −=E e

Ii0 ( )i

Ii t k x

zE e ω −=η

H e

I( )r r0

i t k xyE e ω +=E e

Ir0 ( )r

Ii t k x

zE e ω += −η

H e

II( )t t0

i t k xyE e ω −=E e

IIt0 ( )t

IIi t k x

zE e ω −=η

H e

I medium II medium

Boundary conditions:

( 0 ) ( 0 )y yE x E x− += = = i0 r0 t0E E E+ =

( 0 ) ( 0 )z zH x H x− += = = i0 r0 t0H H H+ =

i0 r0 t0

I I II

E E E− =

η η η

r0 II Ii0 II I

ERE

η − η= =

η + η

t0 IIi0 II I

2ETE

η= =

η + η

Reflection from Conductors

x

y

incident

reflected transmitted“diffuse” wave

I dielectric II conductor

1 0f

δ = ≈π μσ

0II I

in

ηωμη = << η =

σ

II III I

1R η − η= ≈ −

η + η

• negligible penetration

• almost perfect reflection with phase reversal

21

Axial Skin Effect

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Normalized Depth, x / δ

Nor

mal

ized

Dep

th P

rofil

e, F

magnitudereal part

0 ( ) i tyE F x e ω=E e

0 ( ) i t zH F x e ω=H e


/ /( ) x i xF x e e− δ − δ=

1f

δ =π μσ

x

y

propagating wave diffuse wave

dielectric (air) conductor

Transverse Skin Effect

0 0( )zE E J k r=

1f

δ =π μσ

2k i= − ωμσ1 ik = −δ δ

012 ( )

k IEa J k a

=πσ

Jn nth-order Bessel functionof the first kind

02 1

( )( )2 ( )z

k a J k rIJ rJ k aa

=π

z

r

current density

conductor rod

current, I 2a

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1Normalized Radius, r/a

Nor

mal

ized

Cur

rent

Den

sity

, J/J

DC

a/δ = 1a/δ = 3a/δ = 10

magnitude, DC 2IJa

=π

22

Transverse Skin Effect

z

r

current density

conductor rod

current, I 2a

0.1

1

10

100

0.01 0.1 1 10 100

Normalized Radius, a/δ

Nor

mal

ized

Res

ista

nce,

R/R

0

R ∝ ω

0R R≈

VZ R i XI

= = +

0 2RA a

= ρ =σπ

0

1

( )( )2 ( )

JGJ

ξ ξξ =

ξ

0 ( )Z R G k a=

/lim (1 )

2a

aG iδ→∞

= +δ

/lim

2aR

aδ→∞=

σ π δ

23

2 Eddy Current Theory

2.1 Eddy Current Method

2.2 Impedance Measurements

2.3 Impedance Diagrams

2.4 Test Coil Impedance

2.5 Field Distributions

2.1 Eddy Current Method

24

Eddy Current Penetration Depth0 ( ) i t

yE F x e ω=E e

0 ( ) i t zH F x e ω=H e


/ /( ) x i xF x e e− δ − δ=

aluminum (σ = 26.7 × 106 S/m or 46 %IACS)

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Depth [mm]

Re

F

f = 0.05 MHzf = 0.2 MHzf = 1 MHz

f = 0.05 MHzf = 0.2 MHzf = 1 MHz

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Depth [mm]

| F |

1f

δ =π μσ

Eddy Currents, Lenz’s Law

conducting specimen

eddy currents

probe coil

magnetic field

s p s( )dVdt

= − Φ − Φ

p p∇× =H J

s p s( )t

∂∇× = −μ −

∂E H H

s s= σJ E

p p pN IΦ = μ Λ

s sI V∝ σ

s s sIΦ = μ Λs s∇× =H J

secondary(eddy) current

(excitation) currentprimarymagnetic flux

primary

magnetic fluxsecondary

p p s( )dV Ndt

= − Φ − Φ

pprobe

p( , , , , ... )

VZ

I= ω σ μ

25

2.2 Impedance Measurements

Impedance Measurements

pI p

e

( )( )

( )V

K ZI

ωω = =

ωIe VpZp

Ve

Ze

VpZp

Voltage divider:

Current generator:

Iep p

Ve e p

( )( )

( )V Z

KV Z Z

ωω = =

ω +

Ve p

V

( )1 ( )

K Z ZK

ω=

− ω

26

Resonance

Ve

R

L VoC

0

0.2

0.4

0.6

0.8

1

0 1 2 3Normalized Frequency, ω/Ω

Tran

sfer

Fun

ctio

n, |

K|

Q = 2Q = 5Q = 10

p 2( )1

i LZLC

ωω =

− ω

po

e p

( )( )( )( ) ( )

ZVKV R Z

ωωω = =

ω + ω

2/( )

1 /i L RK

i L R LCω

ω =+ ω − ω

2 2( )

1 /

iQK i

Q

ωΩω =

ω+ − ω ΩΩ

1LC

Ω =

C RQ R R CL L

= = = ΩΩ

o 211

4Qω = Ω −

Wheatstone Bridge

32 2e 1 2 4 3

( )( )( )

ZV ZK GV Z Z Z Z

⎛ ⎞ωω = = −⎜ ⎟ω + +⎝ ⎠

VeV2

Z1 Z4

Z2 Z3

+_ G

322

1 40 if ZZV

Z Z= =

1 4 0Z Z R= =

*2 cZ i L R= ω +

3 c cZ i L R= ω +

R0 reference resistanceLc reference (dummy) coil inductanceRc reference coil resistanceL* complex probe coil inductance

2 3 (1 )Z Z= + ξ

probe coil reference coil

3 3

0 3 0 3

(1 )( )(1 )

Z ZK GR Z R Z

⎛ ⎞+ ξω = −⎜ ⎟+ + ξ +⎝ ⎠

0( ) ( )K G Kω ≈ ω ξ

3 00 2

0 3( )

( )Z RK

R Zω =

+

27

Impedance Bandwidth

3 c cZ i L R= ω +

R0 = 100 Ω, Rc = 10 Ω

0( ) ( )K G Kω ≈ ω ξ

3 00 2

0 3( )

( )Z RK

R Zω =

+

0 1 2 30

0.1

0.2

0.3

0.4

0.5

Frequency [MHz]

Tran

sfer

Fun

ctio

n, |

K0

|

Lc = 100 µHLc = 20 µHLc = 10 µH

c 00 2

c 0

/( )1 ( / )

L RKL R

ωω ≈

+ ω

3 cZ i L≈ ω

0p

c

RL

ω =

0 p1( )2

K ω ≈

02

c

2 RL

ω =

0 1,22( )5

K ω ≈

01

c2RL

ω =

21

4ω=

ω

2 1c 2 1

62 or 120%5r

BB ω − ω= = =

ω ω + ω

( , , , ,...)ξ = ξ ω σ μ

2.3 Impedance Diagrams

28

Examples of Impedance Diagrams

Im(Z)

Re(Z)

L

C

Im(Z)

Re(Z)0

Ω-

Ω+

∞

L

C

R 0

Ω-

Ω+

∞R

Im(Z)

Re(Z)

R

L

C

0 Ω∞ R

Im(Z)

Re(Z)

R2

L

C

0 Ω∞ R1 R1+R2

R1

Magnetic Coupling

12 2122 11

Φ Φ= = κ

Φ Φ

2 2 21 22( )dV Ndt

= Φ + Φ

1 1 11 12( )dV Ndt

= Φ + Φ

1 11 12 1

2 21 22 2

V L L Ii

V L L I⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ω⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

12 21 11 22L L L L= = κ

221 11

1

NL LN

= κ 112 22

2

NL LN

= κ

1 1121 11

1

I LN

Φ = κΦ = κ 2 2212 22

2

I LN

Φ = κΦ = κ

1 1111

1

I LN

Φ = 2 2222

2

I LN

Φ =

I1

N1 N2 V2

Φ11

V1

I2

Φ22Φ12 Φ21,

V1 V2L , L , L11 12 22

I1 I2

29

Probe Coil Impedance

e 22222n

e 22 e 22

R i LLZ iR i L R i L

− ωω= + κ

+ ω − ω

2 222 e 222 2n 2 2 2 2 2 2e e22 22

(1 )LL RZ i

R L R Lωω

= κ + − κ+ ω + ω

V2V1

I1 I2

L , L , L11 12 22 Re

2 2 e 12 1 22 2V I R i L I i L I= − = ω + ω

122 1

e 22

i LI IR i L

− ω=

+ ω

1 11 1 12 2V i L I i L I= ω + ω

2 212

1 11 1e 22

( )L

V i L IR i L

ω= ω +

+ ω

2 212

coil 11e 22

LZ i L

R i Lω

= ω ++ ω

222n22e

LZ iR i L

ω= + κ

+ ω

1 11 12 1

2 12 22 2

V L L Ii

V L L I⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ω⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1coil

1

VZI

=

coiln

11(1 )ZZ i

L= = + ξ

ω

coil ref [1 ( , , )]Z Z= + ξ ω σ

ref 11Z i L≈ ω

2 211 2212L L L= κ

( )κ = κ

Impedance Diagram22 eL Rζ = ω /

2n n 2Re

1R Z ζ

= = κ+ ζ

22

n n 2Im 11

X Z ζ= = − κ

+ ζ

n n0 0

lim 0 and lim 1R Xω→ ω→

= =

2n nlim 0 and lim 1R Xω→∞ ω→∞

= = − κ

2 2n n( 1) and ( 1) 1

2 2R Xκ κ

ζ = = ζ = = −

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

κ = 0.6κ = 0.8κ = 0.9

Re=10 Ω

Re=5 Ω

Re=30 Ω

22 e e3 H, = 1 MHz, / 10%L f R R= μ Δ = lift-off trajectories are straight:

n n1X R= − ζ

conductivity trajectories are semi-circles

2 22 22n n 1

2 2R X

⎛ ⎞ ⎛ ⎞κ κ+ − + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

30

Electric Noise versus Lift-off Variation

0.32

0.34

0.36

0.38

0.40

0.42

0.28 0.3 0.32 0.34 0.36 0.38“Horizontal” Impedance Component

“Ver

tical

”Im

peda

nce

Com

pone

nt

0.32

0.34

0.36

0.38

0.40

0.42

0.28 0.3 0.32 0.34 0.36 0.38Normalized Resistance

Nor

mal

ized

Rea

ctan

ce lift-offlift-off

“physical” coordinates rotated coordinates

nZ ⊥ΔnZΔ

Conductivity Sensitivity, Gauge Factor

22 e e3 H, = 1 MHz, 10 , 1L f R R= μ = Ω Δ = ± Ω

nnorm

e e/Z

FR R

⊥Δ=

Δn

abse e/Z

FR RΔ

=Δ

0 (1 )R R F= + ε //

R RF Δ=

Δ

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.2 0.4 0.6 0.8 1Frequency [MHz]

Gau

ge F

acto

r, F

absolute

normal0.32

0.34

0.36

0.38

0.40

0.42

0.28 0.3 0.32 0.34 0.36 0.38Normalized Resistance

Nor

mal

ized

Rea

ctan

ce lift-off

nZ ⊥Δ

nZΔ

31

Conductivity and Lift-off Trajectories

lift-off trajectories are not straightconductivity trajectories are not semi-circles

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ized

Rea

ctan

ce

κ

lift-off

conductivity

eLRA

≈σ

( )κ ≈ κ

e ( )LR

A≈

σ σ( , )κ ≈ κ σ finite probe size

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ized

Rea

ctan

ce

κlift-off

conductivity

2.4 Test Coil Impedance

32

Air-core Probe Coils

single turn L = a L = 3 a

center 2IHa

=

24 rI dd

r= ×

πH e e

a coil radiusL coil length

encd I=∫ H si

center/lim

L a

N IHL→∞

=

2axis 2 2 3/ 22( )

I aHa z

=+

Infinitely Long Solenoid Coil

encd I=∫ H si

sJ n I=

1 2( ) ( ) 0z zL H r L H r− =

for outside loops (r1,2 > a)

0zH =

1 2( ) ( ) 0z zL H r L H r− =

for inside loops (r1,2 < a)

constantzH =

1 2 s( ) ( )z zL H r L H r L J− =

1 s( )zIH r J n I NL

= = =

for encircling loops (r1 < a < r2)

inside loop outside loopencircling

2a

L

+ Js_ Js

z

33

Magnetic Field of an Infinite Solenoid with Conducting Core

in the air gap (b < r < a) Hz = Js

in the core (0 < r < b) Hz = H1 J0(kr)

Jn nth-order Bessel function of the first kind

s1

0( )J

HJ k b

=

+ Js_ Js

2 a2 b

z

0s

0

( )( )z

J k rH J

J k b=

2 2( )k∇ + =H 0 2k i= − ωμσ1 ik = −δ δ

22

21 0zk Hr rr

⎛ ⎞∂ ∂+ + =⎜ ⎟∂∂⎝ ⎠

2 2s

02 ( ) ( )

bzH r r dr a b JΦ = πμ + πμ −∫

z zB dA H dAΦ = = μ∫∫ ∫∫

Magnetic Flux of an Infinite Solenoid with Conducting Core

+ Js_ Js

2 a2 b

z0

s0

( )( )

( )zJ k r

H r JJ k b

=

( )z zH H r=

szH J=

0zH =

2 2s 0

0 0

2[ ( ) ]( )

bJ J k r r dr a b

J k bΦ = πμ + −∫

0 1( ) ( )J d Jξ ξ ξ = ξ ξ∫

1 2 2s

0

2 ( )[ ]

( )b J k b

J a bk J k b

Φ = πμ + −

1

0

2 ( )( )

( )J

gJ

ξξ =

ξ ξ

2 2s [1 ( )]J a b g k bΦ = πμ − −

34

For an empty solenoid (b = 0):

Normalized impedance:

1 1 1, ,s LJ n I V i V NV n LV= = ωΦ = =

1 2 2 2 2s

[1 ( )]LV VZ n L i a b g k b n L

I J= = = ωπμ − −

2 2e eZ i a n L i X= ωπμ =

22

2 is called fill-factor ( lift-off)ba

κ = ≈

2n

e1 [1 ( )]ZZ i g k b

X= = − κ −

2 2s [1 ( )]J a b g k bΦ = πμ − −

Impedance of an Infinite Solenoid with Conducting Core

Resistance and Reactance of an Infinite Solenoid with Conducting Core

2n n n1 [1 ( )]Z i g k b R i X= − κ − = +

0 Re ( ) 1g k b≤ ≤ 0.4 Im ( ) 0g k b− ≤ ≤

2n Im ( )R g k b= −κ 2

n 1 [1 Re ( )]X g k b= − κ −

n n1 RX m= −Re ( ) 1

Im ( )g k bm

g k b−

=

1 ik = −δ δ

(1 ) bk b iξ = = −δ

22 2 bi⎛ ⎞ξ = − ⎜ ⎟δ⎝ ⎠

0.01 0.1 1 10 100 1000-0.4-0.20.00.20.40.60.81.01.2

Normalized Radius, b/δ

g-fu

nctio

n

real partimaginary part

35

Effect of Changing Coil Radius

a (changes)

b (constant)

lift-off

ba

κ =

Normalized Resistance

Nor

mal

ized

Rea

ctan

ce

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

b/δ = 1

3

5

1020

2

κ = 1

0.9

0.8

0.7

a

lift-off

ω

2n 1 [1 ( )]Z i g k b= − κ −

Effect of Changing Core Radius

b (changing)

a (constant)lift-off

2n 1 n 2 n1 R RX m m≈ − −

ba

κ =

n 1 21 0

1, where ( )2

aa

ω δω = ω = =

ω σμ


Nor

mal

ized

Rea

ctan

ce

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

100400

9

25

ω

ωn = 4

κ = 1

0.9

0.8

0.7

b

lift-off

2n 1 [1 ( )]Z i g k b= − κ −

36

Permeability


Nor

mal

ized

Rea

ctan

ce

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2

ωn = 0.6

1.5

1

2

3

1

µr = 4

µ ω

0.8ba

κ = =

n1

ωω =

ω

2 2r 0 0 s

02 ( ) ( )

bzH r r dr a b JΦ = πμ μ + πμ −∫

2n r1 [1 ( )]Z i g bk= − κ − μ

1 20 r

1 ( )2

aa

δω = =

σμ μ

Solid Rod versus Tube2 2 2

0 3 r 0 0 s2 ( ) ( )b

zc

c H H r r dr a b JΦ = πμ + πμ μ + πμ −∫

1 0 2 0( ) ( )zH H J k r H Y k r= +

1 0 2 0 sBC1: ( ) ( )H J k b H Y k b J+ =

1 0 2 0 3BC2: ( ) ( )H J k c H Y k c H+ =

1 1 2 1 3BC3: ( ) ( )2

k cH J k c H Y k c H+ =

b

a

1 1 2 1[ ( ) ( )] ( )k H J k c H Y k c E cϕ− + =σ

∇× = = σH J E

zH Er ϕ

∂− = σ

∂

20 3 ( )2i H c E c cϕωμ π = π

solid rod

BC1: continuity of Hz at r = b

tube

BC1: continuity of Hz at r = b

BC2: continuity of Hz at r = c

BC3: continuity of Eφ at r = c

b

a

c( )z zH H r=

szH J=

0zH =

3zH H=

37

Solid Rod versus Tube

b

a

c

1,b ca b

κ = = η =

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6Normalized Resistance

very thin

solid rod

tubeNor

mal

ized

Rea

ctan

cethick tube

σ1

σ2

σ1

σ2

Wall Thickness

b

a

c

1,b ca b

κ = = η =

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

η = 0solid rod

b/δ = 3

b/δ = 2


Nor

mal

ized

Rea

ctan

ce

b/δ = 5

b/δ = 10

b/δ = 20 η ≈ 1thin tube

η = 0.2η = 0.4η = 0.6η = 0.8

38

Wall Thickness versus Fill Factor

b

a

c

,b ca b

κ = η =

0

0.2

0.4

0.6

0.8

1


Nor

mal

ized

Rea

ctan

ce

solid rodκ = 0.95, η = 0

solid rodκ = 1, η = 0

thin tubeκ = 1, η = 0.99

thin tubeκ = 0.95, η = 0.99

Clad Rod

b

a

c

2 2core core clad clad 0 s

02 ( ) 2 ( ) ( )

c b

cH r r dr H r r dr a b JΦ = πμ + πμ + πμ −∫ ∫

clad 1 0 clad 2 0 clad( ) ( )H H J k r H Y k r c r b= + ≤ <

core 3 0 core( ) 0H H J k r r c= ≤ <

0

0.2

0.4

0.6

0.8

1


Nor

mal

ized

Rea

ctan

ce

copper claddingon brass coresolid

copper rod

solidbrass rodbrass cladding

on copper core

d

master curve forsolid rod

d

thin wall

lower fill factor

clad

core, ,b c

a bσ

κ = η = Σ =σ

(1 )d b c b= − = − η

39

2D Axisymmetric Models

b

a

c

2ao

2ai

t

h

ℓ

short solenoid (2D)

↓long solenoid (1D)

↓thin-wall long solenoid (≈0D)

↓coupled coils (0D)

pancake coil (2D)

o

i1( ) ( )

a

aI x J x dx

α

αα = ∫

2 202 2 6

0o i

( ) ( )( )

i N IZ f dh a a

∞ωπμ α= α α∫

− α

r 1( ) 2r 1

( ) 2( 1) [ ]h hf h e e e−α −α + −α αμ −αα = α + − + −

αμ +α2 2 2 2

r 01 k iα = α − = α + ωμ μ σ

Dodd and Deeds. J. Appl. Phys. (1968)

Flat Pancake Coil (2D)

0

0.05

0.1

0.15

0.2

0.1 1 10 100Frequency [MHz]

(Nor

mal

) Gau

ge F

acto

r

4 mm2 mm1 mm

coil diameter

o iM 2

12

a aa f

a+

= = δ ⇒ =π σμ

a0 = 1 mm, ai = 0.5 mm, h = 0.05 mm, σ = 1.5 %IACS, μ = μ0

0

0.2

0.4

0.6

0.8

1


Nor

mal

ized

Rea

ctan

ce

0 mm

0.05 mm

0.1 mm

lift-off

frequency

fM

40

2.5 Field Distributions

Field Distributions

air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm), in Ti-6Al-4V (σ = 1 %IACS)

10 Hz

10 kHz

1 MHz

10 MHz

1 mm

magnetic field2 2r zH H H= +

electric field Eθ

(eddy current density)

41

Axial Penetration Depth air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V

Axi

al P

enet

ratio

n D

epth

, δ a

[m

m]

10-2

10-1

100

101

Frequency [MHz] 10-5 10-4 10-3 10-2 10-1 100 101 102

standard

actual1f

δ =π σμ

ai

i o11/e point below the surface at ( )2

r a a a= = +

1 22a a≈

Radial Spread air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V

Rad

ial S

prea

d, a

s[m

m]

Frequency [MHz] 10-5 10-4 10-3 10-2 10-1 100 101 102

analytical

finite element

0.8

1.2

1.6

2.0

1.0

1.4

1.8

1/e point from the axis at the surface ( 0)z =

2 o1.2a a≈

42

Radial Penetration Depth air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V

Rad

ial P

enet

ratio

n D

epth

, δr

[mm

]

10-2

10-1

100

101

Frequency [MHz] 10-5 10-4 10-3 10-2 10-1 100 101 102

standard

actual1f

δ =π σμ

r s 2a aδ = −

2 o1.2a a≈

Lateral Resolution ferrite-core pancake coil (ai = 0.625 mm, ao = 1.25 mm, h = 3 mm) in Ti-6Al-4V

1.0

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

experimental

FE prediction

Rad

ial S

prea

d, a

s[m

m]

Frequency [MHz] 10-2 10-1 100 101

43

3 Eddy Current NDE

3.1 Inspection Techniques

3.2 Instrumentation

3.3 Typical Applications

3.4 Special Example

3.1 Inspection Techniques

44

Coil Configurationsvoltmeter

testpiece

oscillator

excitationcoil

sensing coil

~

voltmeter

testpiece

oscillator

coil

Zo

~

Hall or GMR detector

voltmeter

testpiece

oscillator

excitationcoil

~~

differential coils

coaxial rotatedparallel

Remote-Field Eddy Current Inspection

Remote Field Remote FieldNear Field

exciter coilferromagnetic pipe sensing coil

ln(Hz)

z

low frequency operation (10-100 Hz)

Exponentially decaying eddy currents propagating mainly on the outer surface

cause a diffuse magnetic field that leaks both on the outside and the inside of the pipe.

0

1

rfδ =

π μ μ σ

/0

zz zH H e− δ=

45

Time

Sign

al

Main Modes of Operationsingle-frequency time-multiplexed multiple-frequency

frequency-multiplexed multiple-frequencypulsed

Time

Sign

al

Time

Sign

al

Time

Sign

al

excited signal (current) detected signal (voltage)

2Dτ ≈ μσ

Nonlinear Harmonic Analysissingle frequency, linear response

nonlinear harmonic analysis

Time

Sign

al

Time

Sign

al

H

B

ferromagnetic phase(ferrite, martensite, etc.)

46

3.2 Eddy Current Instrumentation

Single-Frequency Operation

low-passfilter

low-passfilter

oscillator driveramplifier

+_

90º phaseshifter

A/Dconverter

display

probe coil(s)

driverimpedances

processorphase

balanceV-gainH-gain

Vr

Vm

Vq

m s s r o q ocos( ), cos( ), sin( )V V t V V t V V t= ω − ϕ = ω = ω

[ ]m r s s o s o s s1cos( ) cos( ) cos( ) cos(2 )2

V V V t V t V V t= ω − ϕ ω = ϕ + ω − ϕ

[ ]m q s s o s o s s1cos( ) sin( ) sin( ) sin(2 )2

V V V t V t V V t= ω − ϕ ω = ϕ + ω − ϕ

o om r s s m q s scos( ), sin( )

2 2V VV V V V V V= ϕ = ϕ

47

Nonlinear Harmonic Operation

low-passfilter

low-passfilter

n divider driveramplifier

+_

90º phaseshifter

A/Dconverter

display

probe coil(s)

driverimpedances

processorphase

balanceV-gainH-gain

oscillatorVr

Vm

Vq

m s1 s1 s2 s2 s3 s3cos( ) cos(2 ) cos(3 ) ...V V t V t V t= ω − ϕ + ω − ϕ + ω − ϕ +

r o cos( )V V n t= ω om r s scos( )

2 n nVV V V= ϕ

q o sin( )V V n t= ω om q s ssin( )

2 n nVV V V= ϕ

Specialized versus General Purpose

≈ 3 minutes for 81 points≈ 50 minutes for 21 pointsmeasurement time

electronicmanualfrequency scanning

≈ 0.05-0.1%≈ 0.1-0.2%relative accuracy

single spiral coilthree pencil probesprobe coil

0.1-80 MHz0.1 – 10 MHzfrequency range*

Agilent 4294A system*Nortec 2000S system

*high-frequency application

48

I1

V2

Φ11

V1

I2

Φ22Φ12 Φ21,

Probe Considerations

V Z I=

*wireZ i L R= ω +

sensitivity

thermal stability

eddy current

ferrite-core coil

high coupling

high coupling

eddy current

air-core coil

high coupling

low coupling

eddy current

flat air-core coilhigh coupling

flexible, low self-capacitance, reproducible, interchangeable, economic, etc.

I

Φ

V

1 11 12 1

2 12 22 2

V Z Z IV Z Z I

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦*

12 12Z i L= ω

topology

3.3 Eddy Current NDE Applications

• conductivity measurement• permeability measurement• metal thickness measurement• coating thickness measurements• flaw detection

49

3.3.1 Conductivity

Conductivity versus Probe Impedance constant frequency

0

0.2

0.4

0.6

0.8

1


Nor

mal

ized

Rea

ctan

ce

StainlessSteel, 304

CopperAluminum, 7075-T6

Titanium, 6Al-4V

Magnesium, A280

Lead

Copper 70%,Nickel 30%

Inconel

Nickel

50

Conductivity versus Alloying and Temper IACS = International Annealed Copper Standard

σIACS = 5.8×107 Ω-1m-1 at 20 °C

ρIACS = 1.7241×10-8 Ωm

20

30

40

50

60C

ondu

ctiv

ity [%

IAC

S]

T3 T4

T6

T0

2014

T4

T6T0

6061

T6

T73T76

T0

70752024

T3 T4

T6

T72T8

T0

Various Aluminum Alloys

Apparent Eddy Current Conductivity

• high accuracy (≤ 0.1 %)

• controlled penetration depth

specimen

eddy currents

probe coil

magnetic field

0

0.2

0.4

0.6

0.8

1.0

0.10 0.2 0.3 0.4 0.5

lift-offcurves

conductivity

curve(frequency)


Nor

mal

ized

Rea

ctan

ce

σ,

σ = σ2

σ = σ1

= 0

= s

1

23

4


Nor

mal

ized

Rea

ctan

ce

51

Lift-Off Curvature

inductive(low frequency)

capacitive(high frequency)

“Horizontal” Component

“Ver

tical

”C

ompo

nent

lift-off

.

conductivity

σ2

σ1

σ

ℓ = s ℓ = 0

“Horizontal” Component

“Ver

tical

”C

ompo

nent

.

conductivity

lift-off

σ2

σ1

σ

ℓ = s ℓ = 0

Inductive Lift-Off Effect

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0


Rel

ativ

e ΔA

ECC

[%] .

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0


Rel

ativ

e ΔA

ECC

[%] .

63.5 μm50.8 μm38.1 μm25.4 μm19.1 μm12.7 μm6.4 μm0.0 μm

-100

1020304050607080


AEC

L [μ

m]

.

-100

1020304050607080


AEC

L [μ

m]

. .

63.5 μm50.8 μm38.1 μm25.4 μm19.1 μm12.7 μm6.4 μm0.0 μm

4 mm diameter 8 mm diameter

1.5 %IACS 1.5 %IACS

52

Instrument Calibration

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0


AEC

C C

hang

e [%

] .

12A Nortec 8A Nortec 4A Nortec 12A Agilent 8A Agilent 4A Agilent 12A UniWest 8A UniWest 4A UniWest 12A Stanford 8A Stanford 4A Stanford

Nortec 2000S, Agilent 4294A, Stanford Research SR844, and UniWest US-450

conductivity spectra comparison on IN718 specimens of different peening intensities

3.3.2 Permeability

53

Magnetic Susceptibility

0

0.2

0.4

0.6

0.8

1.0

0.10 0.2 0.3 0.4 0.5

lift-off

frequency(conductivity)


Nor

mal

ized

Rea

ctan

ce

permeability


Nor

mal

ized

Rea

ctan

ce

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2

2

3

1

µr = 4permeability

moderately high susceptibility low susceptibility

paramagnetic materials with small ferromagnetic phase content

increasing magnetic susceptibility decreases the apparent eddy current conductivity (AECC)

frequency(conductivity)

Magnetic Susceptibility versus Cold Work

10-4

10-3

10-2

10-1

100

101

0 10 20 30 40 50 60Cold Work [%]

Mag

netic

Sus

cept

ibili

ty

SS304L

IN276

IN718

SS305

SS304SS302

IN625

cold work (plastic deformation at room temperature) causesmartensitic (ferromagnetic) phase transformation

in austenitic stainless steels

54

3.3.3 Metal Thickness

Thickness versus Normalized Impedance

thickness loss due to corrosion, erosion, etc.

probe coil

scanning

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

thickplate


Nor

mal

ized

Rea

ctan

ce

thinplate

lift-off

thinning

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Depth [mm]

Re

F

f = 0.05 MHzf = 0.2 MHzf = 1 MHz

aluminum (σ = 46 %IACS)

/ /( ) x i xF x e e− δ − δ=

55

Thickness Correction

1.0

1.1

1.2

1.3

1.4

0.1 1 10Frequency [MHz]

Con

duct

ivity

[%IA

CS]

1.0 mm1.5 mm2.0 mm2.5 mm3.0 mm3.5 mm4.0 mm5.0 mm6.0 mm

thickness

Vic-3D simulation, Inconel plates (σ = 1.33 %IACS)

ao = 4.5 mm, ai = 2.25 mm, h = 2.25 mm

3.3.4 Coating Thickness

56

Non-conducting Coating

non-conductingcoating

probe coil, ao

t

d

ℓ

conducting substrate

ao > t, d > δ, AECL = ℓ + t

-100

1020304050607080


AEC

L [μ

m]

-100

1020304050607080


AEC

L [μ

m]

63.5 μm

50.8 μm

38.1 μm

25.4 μm

19.1 μm

12.7 μm

6.4 μm

0 μm

ao = 4 mm, simulatedlift-off:

ao = 4 mm, experimental

Conducting Coating

conductingcoating

probe coil, ao

t

d

ℓ

conducting substrate (µs,σs)

approximate: large transducer, weak perturbation

equivalent depth:

( )e1AECC( )

2 s sf

f

⎛ ⎞≈ σ δ = σ⎜ ⎟⎜ ⎟π μ σ⎝ ⎠

21( ) AECC

4 s sz

z

⎛ ⎞σ ≈ ⎜ ⎟⎜ ⎟π μ σ⎝ ⎠

se 2

δδ =

analytical: Fourier decomposition (Dodd and Deeds)

numerical: finite element, finite difference, volume integral, etc.(Vic-3D, Opera 3D, etc.)

zJe

z = δe

57

Simplistic Inversion of AECC Spectra

AEC

C C

hang

e [%

]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.001 0.1 10 1000

Frequency [MHz]

AEC

C C

hang

e [%

]-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.001 0.1 10 1000

Frequency [MHz]

Depth [mm]

Con

duct

ivity

Cha

nge

[%]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

input profile

inverted from AECC

uniform

Depth [mm]

Con

duct

ivity

Cha

nge

[%]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

input profile

inverted fromAECC

Gaussian

0.254-mm-thick surface layer of 1% excess conductivity

3.3.5 Flaw Detection

58

Impedance Diagram


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

conductivity(frequency)

crackdepth

flawlessmaterialω1

lift-off

Nor

mal

ized

Rea

ctan

ce

ω2

apparent eddy current conductivity (AECC) decreasesapparent eddy current lift-off (AECL) increases

Crack Contrast and Resolution

probe coil

crack

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5Flaw Length [mm]

Nor

mal

ized

AEC

C

semi-circular crack

-10% threshold

detectionthreshold

ao = 1 mm, ai = 0.75 mm, h = 1.5 mm

austenitic stainless steel, σ = 2.5 %IACS, μr = 1

Vic-3D simulation

f = 5 MHz, δ ≈ 0.19 mm

59

Eddy Current Images of Small Fatigue Cracks

Al2024, 0.025-mil crack Ti-6Al-4V, 0.026-mil-crack

0.5” × 0.5”, 2 MHz, 0.060”-diameter coil

probe coil

crack

Crystallographic Texture= σJ E

1 1 1

2 2 2

3 3 3

0 00 00 0

J EJ EJ E

σ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= σ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥σ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

generally anisotropic hexagonal (transversely isotropic)

1 1 1

2 2 2

3 2 3

0 00 00 0

J EJ EJ E

σ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥= σ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥σ⎣ ⎦⎣ ⎦ ⎣ ⎦

cubic (isotropic)

1 1 1

2 1 2

3 1 3

0 00 00 0

J EJ EJ E

σ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥= σ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥σ⎣ ⎦⎣ ⎦ ⎣ ⎦

σ1 conductivity normal to the basal plane

σ2 conductivity in the basal plane

θ polar angle from the normal of the basal plane

σm minimum conductivity in the surface plane

σM maximum conductivity in the surface plane

σa average conductivity in the surface plane2 2a 1 2( ) ½ [ sin (1 cos )]σ θ = σ θ + σ + θ

2 2n 1 2( ) cos sinσ θ = σ θ + σ θ

M 2σ = σ

1 2σ < σ

2 2m 1 2( ) sin cosσ θ = σ θ + σ θ

x1

x3

x2basal plane

θ

surface plane

σnσm

σM

60

Electric “Birefringence” Due to Texture

1.00

1.01

1.02

1.03

1.04

1.05

0 30 60 90 120 150 180Azimuthal Angle [deg]

Con

duct

ivity

[%IA

CS]

highly textured Ti-6Al-4V plate equiaxed GTD-111

1.30

1.32

1.34

1.36

1.38

1.40


Con

duct

ivity

[%IA

CS]

500 kHz, racetrack coil

Grain Noise in Ti-6Al-4V

as-received billet material solution treated and annealed heat-treated, coarse

heat-treated, very coarse heat-treated, large colonies equiaxed beta annealed

1” × 1”, 2 MHz, 0.060”-diameter coil

61

Eddy Current versus Acoustic Microscopy

5 MHz eddy current 40 MHz acoustic

1” × 1”, coarse grained Ti-6Al-4V sample

InhomogeneityAECC Images of Waspaloy and IN100 Specimens

homogeneous IN100

2.2” × 1.1”, 6 MHz

conductivity range ≈1.33-1.34 %IACS

±0.4 % relative variation

inhomogeneous Waspaloy

4.2” × 2.1”, 6 MHz

conductivity range ≈1.38-1.47 %IACS

±3 % relative variation

62

Conductivity Material Noise

1.30

1.32

1.34

1.36

1.38

1.40

1.42

1.44

1.46

1.48

1.50


AEC

C [%

IAC

S]

Spot 1 (1.441 %IACS)



Spot 4 (1.382% IACS)

as-forged Waspaloy

no (average) frequency dependence

Magnetic Susceptibility Material Noise1” × 1”, stainless steel 304

f = 0.1 MHz, ΔAECC ≈ 6.4 %

f = 5 MHz, ΔAECC ≈ 0.8 %

intact

f = 0.1 MHz, ΔAECC ≈ 8.6 %

f = 5 MHz, ΔAECC ≈ 1.2 %

0.51×0.26×0.03 mm3 edm notch

63

3.4 Special Example

Residual Stress Assessment

106102

intact (no residual stress)

with opposite residual stress

Fatigue Life [cycles]104 108

0

500

1000

1500

endurancelimit

service load

life timenatural

life timeincreasedA

ltern

atin

g St

ress

[MPa

]

Residual stresses have numerous origins that are highly variable.Residual stresses relax at service temperatures.

64

Surface-Enhancement TechniquesLow-Plasticity Burnishing (LPB)Shot Peening (SP) Laser Shock Peening (LSP)

Depth [mm]0 0.2 0.4 0.6 1.0 1.2

200

0

-200

-400

-600

-800

-1000

Res

idua

l Stre

ss [M

Pa]

SP Almen 12ASP Almen 4A

LSPLPB

Ti-6Al-4V

0 0.2 0.4 0.6 1.0 1.2Depth [mm]

Col

d W

ork

[%] 40

30

20

10

0

50

SP Almen 12ASP Almen 4A

LSPLPB

Ti-6Al-4V

Piezoresistive Effect

Electroelastic Tensor:

1 0 11 12 12 1

2 0 12 11 12 2

3 0 12 12 11 3

/ // // /

EEE

Δσ σ κ κ κ τ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥Δσ σ = κ κ κ τ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥Δσ σ κ κ κ τ⎣ ⎦⎣ ⎦ ⎣ ⎦

11 120/

/a

ipip E

Δσ ση = = κ + κ

τ

Isotropic Plane-Stress ( and ) :1 2 ipτ = τ = τ 3 0τ =

parallel, normal, circular

F F

δ

Adiabatic Electroelastic Coefficients:*11 11 thκ = κ + κ*12 1 2 thκ = κ + κ

-40-20

020406080

Time [1 s/div]

Axi

al S

tress

[ksi

]

Time [1 s/div]1.3971.3981.399

1.41.4011.4021.403

Con

duct

ivity

[%IA

CS]

IN 718, parallel

65

Material Types

parallel

-0.004

-0.002

0

0.002

0.004

-0.001 0 0.001 0.002τua / E

Δσ

/ σ0

normal

Copper

Ti-6Al-4V

parallel

-0.004

-0.002

0

0.002

0.004

-0.002 0 0.002 0.004τua / E

Δσ

/ σ0

normalparallel

-0.004

-0.002

0

0.002

0.004

-0.001 0 0.001 0.002τua / E

Δσ

/ σ0

normal

Al 2024

parallel

-0.004

-0.002

0

0.002

0.004

-0.001 0 0.001 0.002τua / E

Δσ

/ σ0

normal

Al 7075

Waspaloy

parallel

-0.004

-0.002

0

0.002

0.004

-0.002 0 0.002 0.004τua / E

Δσ

/ σ0

normal

IN718

parallel

-0.004

-0.002

0

0.002

0.004

-0.002 0 0.002 0.004τua / E

Δσ

/ σ0

normal

XRD and AECC Measurements

-2000

-1500

-1000

-500

0

500

0 0.2 0.4 0.6 0.8Depth [mm]

Res

idua

l Stre

ss [M

Pa]

Almen 4AAlmen 8AAlmen 12AAlmen 16A

-1

0

1

2

3


Con

duct

ivity

Cha

nge

[%] Almen 4A

Almen 8AAlmen 12AAlmen 16A

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8

Col

d W

ork

[%]


Depth [mm]

before (solid circles) and after full relaxation for 24 hrs at 900 °C (empty circles)

-2000

-1500

-1000

-500

0

500

0 0.2 0.4 0.6 0.8Depth [mm]

Res

idua

l Stre

ss [M

Pa]


0

10

20

30

40

50

0 0.2 0.4 0.6 0.8

Col

d W

ork

[%]


Depth [mm]

-1

0

1

2

3


Con

duct

ivity

Cha

nge

[%] Almen 4A

Almen 8AAlmen 12AAlmen 16A

Waspaloy

66

Thermal Stress Relaxation in WaspaloyWaspaloy, Almen 8A, repeated 24-hour heat treatments at increasing temperatures

0.1 0.16 0.25 0.4 0.63 1 1.6 2.5 4 6.3 10

Frequency [MHz]

0

0.1

0.2

0.3

0.4

0.5

0.6

App

aren

t Con

duct

ivity

Cha

nge

[% ] intact

300 °C350 °C400 °C450 °C500 °C550 °C600 °C650 °C700 °C750 °C800 °C850 °C900 °C

The excess apparent conductivity gradually vanishes during thermal relaxation!

XRD versus Eddy Current

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.01 0.1 1 10Frequency [MHz]

AEC

C C

hang

e [%

]

eddy current

0.0 0.5 1.0 1.5Depth [mm]

Col

d W

ork

[%]

.

0

5

10

15

20XRD

.

-1400

-1200

-1000

-800

-600

-400

-200

0

200

0.0 0.5 1.0 1.5Depth [mm]

Res

idua

l Stre

ss [M

Pa]

eddy currentXRD

inversion of measured AECC in low-plasticity burnished Waspaloy

67

0

10

20

30

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Depth [mm]

Col

d W

ork

[%]

.

Almen 4A (XRD) Almen 8A (XRD) Almen 12A (XRD)

-1800-1600-1400-1200-1000

-800-600-400-200

0200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Depth [mm]

Res

idua

l Stre

ss [M

Pa]

.

Almen 4A (AECC) Almen 8A (AECC) Almen 12A (AECC) Almen 4A (XRD) Almen 8A (XRD) Almen 12A (XRD)

≈ 50 MHz

XRD versus High-Frequency Eddy Currentshot peened IN100 specimens of Almen 4A, 8A and 12A peening intensity levels

68

4 Magnetic NDE

4.1 Magnetic Properties

4.2 Magnetic Measurements

4.3 Magnetic Materials Characterization

4.4 Magnetic Flaw Detection

4.1 Magnetic Properties

69

Magnetization

M magnetizationV volumeχ magnetic susceptibility

H magnetic field


μ0 permeability of free space

μr relative permeability

pm magnetic dipole momentN number of turnsI currentA encircled vector area

m N I=p A

+I -I

mV

∑=pM

= χM H

0 0 r( )= μ + = μ μB H M H

r 1μ = + χ

m12

Q= ×p R v

Q charge

v velocity

R radius vector

Classification of Magnetic MaterialsDiamagnetism:

μr < 1no remanenceorbit distortion

e.g., copper, mercury, gold, zinc

Paramagnetism:μr > 1

no remanenceorbit and spin alignment

e.g., aluminum, titanium, platinum

Ferromagnetism:μr >> 1

remanence, coercivity, hysteresisself-amplifying paramagnetism

Curie temperaturee.g., iron, nickel, cobalt

70

Diamagnetism

pm magnetic dipole momentpspin electron spinporb electron orbital motionN number of turns

I current

A encircled area

e charge of proton

τ orbital period

r orbital radius

v orbital velocity

Ei induced electric field

Fe decelerating electric force

m mass of electron

n dipoles within unit volume

χ magnetic susceptibility

vQ

Fm

B

vQFe

B

ieF e E=

mF ev B=

m orb spin= +p p p

2orb 2

Q A e r vp N I Ar

π= = = −

τ π

orb 2er vp = −

2 22 2 0orb 4 4

e re rp B Hm m

μΔ = − = −

ei2 2 Fd r E r

dt eΦ

− = π = − π

2d m dvrdt e dtΦ

= π

2 2 mB r r ve

π = π Δ

2erv Bm

Δ =

- χ ≈ 1-10 ppm

2 20orb 4

e rnm

μχ = −

Weak Paramagnetism, Curie Lawm orb spin= +p p p



Fm magnetic force

Tm twisting moment or torque

Um potential energy of the dipole

kB Boltzmann constant

T absolute temperature

n dipoles within unit volume


m m= ×T p B

m mU = −p Bi

m m90 90

( ) sinU T d p B dθ θ

= θ θ = θ θ∫ ∫

m m cosU p B= − θ

m m sinT p B= θ

m m0

Bm( )U U

k Tp U e−

−=

20

B3n mM C

H k T Tμ

χ = = =

Curie Law:

χ ≈ 5-50 ppm

+I

-Ipm

Fm

B

Fm

Tm

θ

71

Strong Paramagnetism, Curie-Weiss Law:

t iH H H H M= + = + α

tCM HT

=

t i

M M MM TH H H MC

χ = = =− − α

Curie-Weiss law:C

CT T

χ =−

MH

χ =

M magnetization

H exciting magnetic field


C material constant

T absolute temperature

Ht total magnetic field

Hi interaction field

α material factor

TC Curie temperature

Curie law:CM HT

≈

CT C

χ =− α

CT

χ ≈

Ferromagnetism(i) magnetic polarization is produced by collective action of

similarly oriented spins within magnetic domains

(ii) very high permeability

(iii) magnetic hysteresis

(v) remnant magnetic polarization (remanence)

(vi) coercive magnetic field (coercivity)

(iv) depolarization above the (magnetic) Curie temperature

H

B

Br

Hc

first magnetization

72

Spontaneous Magnetization

N N N N

S S S S

N S N S

S N S N

N N S S

S S N N

[100]

[010] “easy” magnetic axis

[001]

[110]

[111]

total internal wall externalU U U U= + +

Magnetic Domains in Single Crystalseasy magnetic axes

H = 0

H

H

H

1 demagnetization(spontaneous magnetization)

4 technical saturation

3 “knee” of the magnetization curve

2 partial magnetization

domain wallmovement

irreversiblerotation

reversiblerotation

H

B

1

2

354

5 full saturation(no precession)

thermal precession not shown

73

4.2 Magnetic Measurements

Magnetic Sensors

10-2

10-1

100

101

102

103

104

105

0 5 10 15 20 25Frequency [Hz]

Flux

Den

sity

[pT/

Hz1/

2 ]

Hall

GMR

SDP

fluxgate

SQUID

noise threshold

axialdV N i N ABdtΦ

= − = − ωcoil:

74

Hall Detector

I I

a

b

xyz

x x

Bz

VH

FmFe

( )Q= + ×F E v B

( ) 0y y x zF e E v B= − + =

Hy

VEa

=

x xI en ab v= −

Hx

y x z zIV a E av B B

enb= = − =

HH

xz

R IV Bb

=

H1R

en=

Fluxgate

Iexc

Vsens

B1

B2

B

hard magnetic cores

high-frequencyexcitation

low-frequency or dcexternal magnetic field

B1 + B2

B2

B1

B1 + B2

B2

B1

B = 0 B ≠ 0

t

t

t

t

t

t

H

B

sensing voltage(to be low-pass filtered)

75

Vibrating-Sample Magnetometer

Vsens B0

vibration (ω)

0 sin( )d d t= ω

1 0 0( ) [ sin( )]t A B M tΦ = + μ κ ω

2 0 0( ) [ sin( )]t A B M tΦ = − μ κ ω

1 2sens( )V t N N

t t∂Φ ∂Φ

= − +∂ ∂

0

0

BM = χ

μ

sens 0( ) 2 cos( )V t N A B t= − ωχ κ ω

B0 bias magnetic flux density

M magnetization


µ0 permeability of free space

d specimen displacement

d0 specimen amplitude

ω angular frequency

t time

κ geometrical coupling factor

A coil cross section

Φ1,2 flux in coil 1 and 2

N number of turns

Vsens sensing voltage

Faraday Balance

Um magnetic potential energy



M magnetization

V volume

Ug gravitational potential energy

U total potential energy

h height

W actual weight

W’ apparent weight


H magnetic field


for a single dipole:

for a given magnetized volume:

precision scale

specimen

W’ = W - Fm

electromagnet

spacerh

m mU = −p Bi

g mU U U= +

' dU dBW W M Vdh dh

= = −

mU M V B= −

U W h M V B= −

M H= χ

200'

2VdH dHW W V H

dh dhμ

− = − μ χ = − χ

76

4.3 Magnetic Materials Characterization

Magnetic Properties

-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5Magnetic Field [kA/m]

Flux

Den

sity

[Tes

la]

hardened steel

soft iron

0 0( , ) ( , )p pB B H M H M H M= = μ + μferromagnetic materials:

para- and diamagnetic materials: 0 ( )B H M= μ +

M H= χ

0 rB H= μ μ

r 1μ = + χ

77

Initial Magnetizationanhysteretic initial magnetization curve

Flux Density

Differential Permeability

Magnetic FieldFl

ux D

ensi

ty


H magnetic field

M magnetization


µd differential permeability

M0 saturation magnetization

n dipoles per unit volume


ddBdH

μ =

0limH

M M→∞

=

0 ( )B H M= μ +

0 mM n p≤

Retentivity, Coercivity, Hysteresis

Br remanence [Vs/m2]

Mr remnant magnetization


Hc coercive field [A/m]

Hci intrinsic coercivity

U0 magnetic energy density

A hysteresis area [J/m3]

0 ( )B H M= μ +

p( , )M M H M=

technical saturation:

HH

B

Br

Hc

r 0 rB M= μ

c c( ) 0H M H+ =

ci( ) 0M H =

c ciH H≤

0dU B dH=

0U AΔ =

78

Texture, Residual Stress

-2

-1

0

1

2

-300 -200 -100 0 100 200 300Magnetic Field [A/m]

Flux

Den

sity

[T]

σ = 0 MPa B||

B⊥

-2

-1

0

1

2


Flux

Den

sity

[T]

σ = 36 MPa B||

B⊥

-2

-1

0

1

2


Flux

Den

sity

[T]

σ = 183 MPaB||

B⊥

-2

-1

0

1

2


Flux

Den

sity

[T]

σ = 110 MPa B||

B⊥

mild steel (Langman 1985)

Magnetostriction

Induced magnetostriction:

Ms spontaneous magnetization

M0 saturation magnetization

e spontaneous strain within a single domain

ε1,2,3 principal strains

H

123e

ε =

12,3 2 3

eεε = − = −

1 2 eε − ε =

Spontaneous magnetostriction:

easy magnetic axes

H = 0

domains 0M M M= ≤

domain domain1 2,3, 0eε = ε =

volume1,2,3 3

eε =

volume 0M ≈

79

Barkhausen Noise

H = 0

H

domain wallmovementH

B

magnetic field Barkhausen noise

Am

plitu

de

Time

• magnetic Barkhausen noise• acoustic Barkhausen noise

Curie Temperature

ferromagnetic materials (T < TC):

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4T / TC

Ms

/ M0

typical pure metal

typical alloy


C material constant

T temperature

TC Curie temperature

Curie-Weiss law:C

CT T

χ =−

80

4.4 Magnetic Flaw Detection

Magnetic Flux Leakage

Advantages:

fast

inexpensive

large, awkward shaped specimens (particle)

Disadvantages:

material sensitive

poor sensitivity

poor penetration depth

ferromagnetic test piece

sensor

Hall cell, etc.)(small coil,

exciter coil

81

Magnetic Boundary Conditions

xt

medium I

medium II

BIθΙ

boundary

BII

BII,t

BII,n

θΙΙ

BI,n

BI,t

xn

xt

medium I

medium II

HI

θΙ

HII

HII,t

HII,n

θΙΙ

HI,n

HI,t

xn

Ampère's law:

∇× =H J

Gauss' law:

0∇ =Bi

I,n II,nB B= I,t II,tH H=

I I,n II II,nH Hμ = μ I I,n II II,ntan tanH Hθ = θ

I III II

tan tanθ θ=

μ μ

Magnetic Refraction

I III II

tan tanθ θ=

μ μ

µI/µII = 1030

100

0 15 45 60 75 900

15

30

45

60

75

90

30Ferromagnetic Angle, θI [deg]

Non

mag

netic

Ang

le, θ

II[d

eg]

medium I(ferromagnetic)

BI

BIIθΙΙ

θΙ

medium II(air)

medium I(ferromagnetic)

BI

BII

θΙΙ

θΙ

medium II(air)

82

Exciter Magnets

electromagnet

air gap

ferromagnetic core

H d N I MMF= =∫

0 r H AΦ = μ μ

0 rMMF d

AΦ

= ∫μ μ

mMMFR =

Φ

m0 r 0 r

1 1 ii i i

dRA A

= ≈ ∑∫μ μ μ μ

H magnetic field

N number of turns

I excitation current

MMF magnetomotive force

Φ magnetic flux

ℓ length of flux line

µ0 µr magnetic permeability

A cross section area

Rm magnetic reluctance

Yoke Excitation

Detection Methods:

• magnetic particle(gravitation, friction, adhesion,cohesion, magnetization)

• magnetic particle with ultraviolet paint

• coil

• Hall detector, GMR sensor

• fluxgate, etc.

Lateral Position

Tang

entia

l Mag

netic

Fie

ld

Lateral Position

Nor

mal

Mag

netic

Fie

ld

electromagnet

crack

N I

magnetometer

83

Subsurface Flaw Detection

H

B

1

2

saturation greatly reduces the differential permeability

crack

low magnetic field

crack

high magnetic field

84

5 Current Field Measurement

5.1 Alternating Current Field Measurement

5.2 Direct Current Potential Drop

5.3 Alternating Current Potential Drop

5.1 Alternating Current Field Measurement

85

Principle of Operation

electric field

magneticflux

density

axial (x)

transverse (y)

normal (z)

galvanic current injection

≈

magnetometer

magnetic injection:

primary ac flux

~~

Bx0

Field Perturbation

magneticflux

density

magnetometer

axial (x)

transverse (y)

normal (z)

axial flaw

cw current

Bz < 0

electriccurrent

Bz > 0

ccw current

axial scanningabove flaw Axial Position

B z[a

.u.]

Axial Position

B x[a

.u.]

B z[a

.u.]

Bx [a.u.]

86

Uniform Field

advantages:

• testing through coatings

• depth information

• limited boundary effects

disadvantages:

• reduced sensitivity

• sensitivity to geometry

• flaw orientation

effect of coating thickness on axial magnetic flux density Bx(ferrous steel, 5 kHz, δ ≈ 0.25 mm, 30-mm-long solenoid)

8

7

6

5

4

3

2

1

00 5 10 15 20

Coating Thickness [mm]

ΔBx

[%]

50 × 5 mm20 × 2 mm20 × 1 mm

slot size

30

25

20

15

10

5

0

Slot Depth [mm]

ΔBx

and

ΔB z

[%]

0 0.5 1 1.5 2 2.5

Bx at 5 kHzBz at 5 kHzBx at 50 kHzBz at 50 kHz

Axial Flaw

8

7

6

5

4

3

2

1

00 10 20 30 40

Slot Depth [mm]

ΔBxm

per 1

mm

Slo

t Dep

th [

%]

40-mm-longsolenoid

12-mm-longsolenoid

rate of increase of the minimum of Bx with slot depth at the center

2-mm-diameter coil, ferrous steel

changes normalized to Bx0

(parallel to B, normal to E)

87

Flaw Orientation

0.17

0.16

0.15

0.14

0.13

0.12

0.11

B x[T

]

0 1 2 3 4 5Scanning time [a. u.]

transverse flaw(normal to B)

axial flaw(normal to E)

0.025

0.020

0.150

0.100

0.05

0

-0.05

B z[T

]

0 1 2 3 4 5Scanning Time [a. u.]

transverse flaw(normal to B)

axial flaw(normal to E)

eddy current mode

magnetic flux mode

Magnetic Flux Mode

electromagnet

crack

N I

magnetometer

Lateral Position

Tang

entia

l Mag

netic

Fie

ld

Lateral Position

Nor

mal

Mag

netic

Fie

ld

88

5.2 Direct Current Potential Drop

Inductive versus Galvanic Coupling

specimen

eddy currents

probe coil

magnetic field

electric current

VI I

injectioncurrent

potential drop

specimen

advantages of galvanic coupling

dc and low-frequency operation

constant coupling (four-point measurement)

awkward shapes

absolute measurements

inherently directional

89

Thin-Plate Approximationcombined electric current and potential field2a

2b

t << a

( ) ( )2

IE r J rr tρ

= ρ =π

( ) ( )2r r

I drV r E r drt r

∞ ∞ρ= =∫ ∫

π

( ) ln const2IV r r

tρ

= − +π

( ) ( )V V V+ −Δ = −

lnI a bVt a bρ +

Δ =π −

[ ]2 ( ) ( )V V a b V a bΔ = − − +

I (+) I (-)V (+) V (-)

I (+) I (-)

V (+) V (-)

Lateral Spread of Current Distribution

( )2

IJ rr t

=π

(0,0) IJat

=π

2 2 2 22(0, )

2

I aJ wa w t a w

=π + +

2 2(0, )( )

I aJ wa w t

=π +

2

(0,0) 2(0, )J

J w=

2 22

22

(0,0) 2(0, )

a wJJ w a

+= =

2w a=

2w I (+) I (-)

V (+) V (-)

x

y

2a

J(0,w)

J(0,0)

90

Thick-Plate Approximation

2( ) ( )2

IE r J rrρ

= ρ =π

2( ) ( )2r r

I drV r E r drr

∞ ∞ρ= =∫ ∫

π

( ) const2IV r

rρ

= +π

t >> a

2a2b

I (+) I (-)V (+) V (-)

≈

combined electric current and potential field

I (+) I (-)V (+) V (-)

( ) ( )V V V+ −Δ = −

1 1IVa b a b

⎡ ⎤ρΔ = −⎢ ⎥π − +⎣ ⎦

[ ]2 ( ) ( )V V a b V a bΔ = − − +

Finite Plate Thickness

2a2b

t

2 2 1/ 2( )2 [ (2 ) ]n

IV rr nt

∞

= −∞

ρ= ∑

π +

2 2 1/ 2

2 2 1/ 2

1[( ) (2 ) ]

1[( ) (2 ) ]

n

IVa b nt

a b nt

∞

= −∞

⎡ρΔ = ∑ ⎢π − +⎣

⎤− ⎥+ + ⎦

I (+) I (-)

V (+) V (-)

n = 0

n = -1

n = +1

n = -2

n = +2

2t

I (+) I (-)V (+) V (-)

91

Resistance versus Thickness

0

1lim lnt

a bt a b→

+Λ =

π −2 2

2limt

ba b→∞

Λ =π −

VRI

Δ= = ρΛ

0.1

1

10

0.01 0.1 1 10 100Normalized Thickness, t / a

Nor

mal

ized

Res

ista

nce,

Λ

finite thicknessthin-plate appr.thick-plate appr.

a = 3b

Crack Detection by DCPDintact specimen

I (+) I (-)V (+) V (-)

( ) ( )0V V V+ −− = Δ

t

I (+) I (-)V (+) V (-)

cracked specimen

( ) ( )cV V V+ −− = Δ

c

1

2

3

0 0.2 0.4 0.6 0.8 1Normalized Crack Depth, c / t

Nor

mal

ized

Pot

entia

l Dro

p, Δ

V c/ Δ

V 0

a / t =0.441.21.8

a = 3b

infinite slot

92

Technical Implementation of DCPD

• low resistance, high current

• thermoelectric effect, pulsed, alternating polarity

• control of penetration via electrode separation

• low sensitivity to near-surface layer

• no sensitivity to permeability

powersupply

polarityswitch

+_

specimen

electrodesVs

+_

5.3 Alternating Current Potential Drop

93

Direct versus Alternating CurrentDCPD

ACPD

• higher resistance, lower current

• no thermoelectric effect

• control of penetration via frequency

• higher sensitivity to near-surface layer

• sensitivity to permeability

Thin-Plate/Thin-Skin Approximation

0lim lnf

V a bI t a b→

Δ ρ +=

π −

2a2b

t << aI (+) I (-)V (+) V (-)

Re lnV a bI T a b

Δ ρ +⎧ ⎫ ≈⎨ ⎬ π −⎩ ⎭

min ,T t≈ δ

fρ

δ =π μ

lim Re lnf

V f a bI a b→∞

Δ ρμ +⎧ ⎫ ∝⎨ ⎬ π −⎩ ⎭

94

Skin Effect in Thin Nonmagnetic Plates

t( )f f tδ = ≈

t 20

1ft

=πμ σ

analytical prediction

a = 20 mm, b = 10 mm, t = 2 mm

100

101

102

103

100 101 102 103 104 105

Frequency [Hz]

Res

ista

nce

[µΩ

]

1 %IACS2 %IACS5 %IACS

10 %IACS20 %IACS50 %IACS

100 %IACS

ft

a = 20 mm, b = 10 mm, σ = 50 %IACS

0.05 mm0.1 mm0.2 mm0.5 mm

1 mm2 mm5 mm

ft100

101

102

103

100 101 102 103 104 105

Frequency [Hz]

Res

ista

nce

[µΩ

]

Skin Effect in Thick Nonmagnetic Plates304 austenitic stainless steel, σ = 2.5 %IACS, experimental

101

102

103

104

100 101 102 103 104 105

Frequency [Hz]

Res

ista

nce

[µΩ

]

50 mm

20 mm

10 mm

6.25 mm

2.5 mm

2 mm

1 mm

0.5 mm

0.2 mm

0.1 mm

0.05 mm

a = 10 mm, b = 7.5 mm

95

Current Distribution in Ferritic Steelf = 0.1 Hz

FE predictions (Sposito et al., 2006)

f = 50 Hz

f = 1 kHz

a = 10 mm, b = 5 mm, t = 38-mm, c = 10 mm (0.5-mm-wide notches, two separated by 5 mm)

Thin-Skin ApproximationVZ R i XI

Δ= = +

c 0 c 0c

0 0

R RK

R− Γ − Γ

= ≈Γ

0 ln a ba b

+Γ =

− c2ln a b c

a b+ +

Γ =−

*0 0R R≈ Γ *

c cR R≈ Γ

0

1

2

1 2 3Electrode Shape Factor, a / b

Elec

trode

Gai

n, Γ

0

2b2a

2b2a

c

0 ln a bRa b

ρ +≈

πδ −2lnc

a b cRa b

ρ + +≈

πδ −

c0 0

1 2limc

cKa b→

≈Γ +

* fR μρ=

π

96

Technical Implementation of ACPD

low-passfilter

low-passfilter

oscillator differentialdriver

+_

90º phaseshifter

A/Dconverter

specimen

electrodesPC

processor

Vr

Vs

Vq

frequency range: 0.5 Hz - 100 kHz

driver current: 10-200 mA

resistance range: 1-10,000 µΩ

common mode rejection: 100-160 dB .

a = 0.160”

b = 0.080”w = 0.054”

2 d = 0.120”

voltagesensing

currentinjection

welding

weldment

d

w

edge weld

clamshellcatalytic converter

Application Example: Weld Penetration

NDE [mil]

Frac

ture

Sur

face

[mils

]

01020304050607080

0 10 20 30 40 50 60 70 80

weld penetration (w)

Weld Penetration [mil]

Res

ista

nce

[µΩ

]

0

50

100

150

200

0 20 40 60 80 100 120

b = 120 mils

80 mils100 mils

electrode separation (b)

97

Application Example: Erosion Monitoring

0 5 10 15 20Time [day]

20

21

22

23

24

25

Tem

pera

ture

[ºC

]

32.0

32.2

32.4

32.6

32.8

33.0

Res

ista

nce

[µΩ

]

0 5 10 15 20Time [day]

20

21

22

23

24

25

Tem

pera

ture

[ºC

]32.0

32.2

32.4

32.6

32.8

33.0

Res

ista

nce

[µΩ

]

erosionerosion

before compensation after compensation

β ≈ 0.001 [1/ºC]

0 0( ) [1 ( )]T T Tρ ≈ ρ + β −

5060708090

100110120130

0 200 400 600 800Temperature [ºC]

Res

istiv

ity [µ

Ωcm

]

301302303304309310316321347403

internalerosion/corrosion

pipe

98

6 Special Methods

6.1 Microwave Techniques

6.2 Dielectric Measurements

6.3 Thermoelectric Measurements

6.1 Microwave Techniques

99

Electromagnetic Spectrum

34 19, 6.63 10 Js, 1.6 10 CE h eV h e− −= ν = ≈ × ≈ ×

microwave

IR light

cosmic rays

X-rays

γ rays

UV light

visiblelight

radio frequency

Frequency [Hz]

10 1081064 10 1014101210 10 1020101816 1022 1024

Energy [eV]

10 10-610-8-10 10 10010-2-4 10 1061042 108 1010

Wavelength [m]

10 1001024 10 10-610-4-2 10 10-1210-10-8 10-14 10-16

typical lattice constant

Electromagnetic WavesPlane waves:

in dielectrics:

( )0

i t k xy y yE E e ω −= =E e e ( )

0i t k x

z z zH H e ω −= =H e e

0

0

E iH i

ωμη = =

σ+ ωε

( )k i i= − ωμ σ + ωε

in conductors:

/ ( / )0

x i t xyE e e− δ ω − δ=E e

/ ( / )0

x i t xzH e e− δ − ω − δ=H e

1 ik = −δ δ

1i iωμ +η = =

σ σδ

1f

δ =π μσ

00

0377μ

η = ≈ Ωε

0 0

0 r nμ η

η = ≈ε ε

( / )0

i t x cyE e ω −=E e

( / )0

i t x czH e ω −=H e

kcω

=

0

0 0

1

r

ccn

= =μ ε ε

80

0 0

1 3 10 m/sc = ≈ ×μ ε

100

Reflection/Transmission between Dielectrics

x

y

incident

reflected transmitted

I dielectric II dielectric

0 0I II

I II,

n nη η

η = η =

• strong penetration

• perceivable reflection

I III II

n nRn n

−=

+

Reflection from Conductors

x

y

incident

reflected transmitted“diffuse” wave

I dielectric II conductor

1 0f

δ = ≈π μσ

0II I

in

ηωμη = << η =

σ

II III I

1R η − η= ≈ −

η + η

• negligible penetration

• almost perfect reflection with phase reversal

101

Far-Field Measurement Configurations

detectorisolatoroscillatorcirculator

hornantenna

specimen

reflection (monostatic radar, pulse-echo)

detector

isolatoroscillator

hornantenna

specimen

transmission (bistatic radar, pitch-catch) scattering (bistatic radar, pitch-catch)

isolatoroscillator

hornantenna

specimen

detector detec

tor

Near-Field Inspection


open-endedwaveguidespecimen stand-off

distance

air backing

foam coreadhesivesubstrateskin laminate

corrosion damage

coating

102

(Qaddoumi et al., 1997)

Microwave Image of Rust Under Paint

40 mm × 40 mm area of rust on a steel plate

24 GHz, 12.5 mm standoff distance, 0.267 mm of paint

60

40

20

06040200 [mm]

[mm]

Lock-in Thermographyglass fiber-reinforced polymer plates (50 × 75 mm2)

(Diener, 1995)


open-endedwaveguidespecimen stand-off

distance

infraredcamera

lock-inamplifier

modulator

microwave raster scan

lock-in thermography(phase image)

150-µm-thickdelamination

bondingdefects

103

6.2 Dielectric Measurements

Fundamentals

t∂

∇× = +∂DH J

t∂

∇× = −∂BE

Maxwell's Equations:

Harmonic solution:

i iωε = σ + ωε

t∂

∇× = σ + ε∂EH E

t∂

∇× = −μ∂HE

i∇× = ωεH E

i∇× = − ωμE H

i σε = ε −

ω

= σJ E

= εD E

= μB H

E electric field

H magnetic field

D electric flux density


J electric current density

σ electric conductivity

ε electric permittivity

µ magnetic permeability

complex electric permittivity

ω angular frequency

t time

ε

104

Electric Polarizatione dQ Q d= =p d e

+Q -Q +Q -Q

E

FeFe

e e= ×T p Ee tQ=F E

E

0= ε +D E P

ee 0V

∑= = χ εpP E

P electric polarization

pe electric dipole moment

V volume

χe electric susceptibility

ε0 permittivity of free space

dipole formation dipole rotation

0 r= ε εD E

r e1ε = + χ

Capacitance

QDA

ACDE

V E

⎫≈ ⎪⎪ ε⎪ ≈⎬= ⎪ε⎪⎪≈ ⎭

Q CV= dQ dVI Cdt dt

= =1V I dtC

= ∫

0 rε = ε ε

Y i C G= ω +

1Zi C

=ω

Y i C= ω

AG = σ

Y i C= ω

( ) '( ) ''( )iε ω = ε ω − ε ω

E

Q A

I

ideal dielectric lossy dielectric

( ) i σε ω = ε −

ω

conducting dielectric

Y i C= ω

AC = ε

''( )tan'( )

D ε ω= δ =

ε ω

105

Complex Electric Permittivity( ) '( ) ''( )iε ω = ε ω − ε ω

frequency [Hz]

Elec

tric

Perm

ittiv

ity [a

. u.]

+

_

ε’

ε’’

_

+dipolar

+

_

+

atomicresonance electronic

resonance

ionic

103 106 109 1012 1015 1018

_

's s0 0lim ( ) lim i

ω→ ω→

σε = ε ω = ε −

ω

Capacitive Probesparallel plate electrodes

sensor with guard electrodes

Vg

basic sensor

Rg

≈

Vm

Im

stray field electrodes

Vg

Rg

≈

Vm

≈Vm

Im×1

buffer

106

Auto-Balancing Bridge

Vg

Rg

≈

Im

H

deviceundertest L

+

_

RrefIm

high-gainoperationalamplifier

2 m refV I R=

1 m dutV I Z=

1dut ref

2

VZ R

V=

1V 2V

dutZ

vectorvoltmeter

vectorvoltmeter

“virtual”ground

Woven Composite

0

10

20

30

40

0.1 1 10 100Frequency [kHz]

Cap

acita

nce

[pF]

.

coateduncoated

0.001

0.01

0.1

1

10

0.1 1 10 100Frequency [kHz]

Con

duct

ance

[μS]

.

coateduncoated

conductive cloth for electric shielding

107

Adhesively Bonded CompositePethrick et al., 2002

0 0.5 1.0 1.5 2.0 2.5Water Uptake [%]

Thic

knes

s Var

iatio

n [%

]

2.5

2.0

1.5

1.0

0.5

0.00 10 20 30 40 50 60 70 80

Time1/2 [hr1/2]

Wat

er U

ptak

e [%

]

2.5

2.0

1.5

1.0

0.5

0.0

intact122 hr580 hr

1,007 hr1,590 hr5,350 hr

Frequency [Hz]

Rel

ativ

e Pe

rmitt

ivity

50

40

30

20

10

010-1 100 101 102 103 104 105 106 107 108 109

Frequency [Hz]

Die

lect

ric L

oss

103

102

101

100

10-1

10-2

10-1 100 101 102 103 104 105 106 107 108 109

intact122 hr580 hr

1,007 hr1,590 hr5,350 hr

6.3 Thermoelectric Measurements

108

Thermoelectric EffectSeebeck, Peltier, and Thomson effect: coupled electric and thermal flux

J electric current density

h thermal flux density

σ electric conductivity (∇T = 0)

κ thermal conductivity (∇V = 0)

V voltage

T temperature

S thermoelectric power

closed-circuit Seebeck effect:

hA

T1 T2

A

B hB

JA

JB

I

open-circuit Seebeck effect:

T1 T2

A

B

hA

hB

JA = 0

JB = 0

VS

T0 T0V+ _

S VS T Tσ σ −∇⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥σ κ −∇⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Jh

V= − σ∇JT= − κ∇h

( )V S T= − σ ∇ + ∇ =J 0 V S T∇ = − ∇

01 2

0 1 2S B A B

TT T

T T TV S dT S dT S dT= + +∫ ∫ ∫

2 2

1 1S A B AB( )

T T

T TV S S dT S dT= − =∫ ∫

Absolute Thermoelectric Power

Temperature [K]0 500 1000 1500

-40

-30

-20

-10

0

10

20

30

Ther

moe

lect

ric P

ower

[µV

/K] W (tungsten)

Mo (molybdenum)

Ag (silver)

Cu (copper)

Au (gold)

Pt (platinum)

Pd (palladium)

2 2

1 1S A B AB( )

T T

T TV S S dT S dT= − =∫ ∫

S AB 2 1( )V S T T≈ −

109

Contact Thermoelectric Tester

Primary Effect:

• chemical composition

Secondary Effects:

• anisotropy, texture

• fatigue, cold work, plasticity, residual stress, etc.

open-circuit Seebeck effect

specimen (A)

electrical heating

“cold” junction “hot” junction

referenceelectrodes

(B)

~~

V+ _

TEP versus Chemical Composition

Ag Content [%]

20

0

-20

-40

-60

Ther

moe

lect

ric P

ower

[µV

/K]

0 20 40 60 80 100

273 K83 K

Ag Content [%]

50

40

30

20

10

0

Elec

tric

Res

istiv

ity [µ

Ωcm

]

0 20 40 60 80 100

293 K4.2 K

palladium-silver binary alloy

(Rudnitskii, 1956)

110

TEP Anisotropy

hexagonal single crystal

Zinc, relative to basal plane

(Rowe and Schroeder, 1970)

Temperature [K]

-3

-2

-1

0

1

2

3

0 50 100 150 200 250 300

perpendicular

parallelTh

erm

oele

ctric

Pow

er [µ

V/K

]

TEP versus Texturecold-worked polycrystalline material

Ti-6Al-4V, relative to cold work direction

(Carreon and Medina, 2006)

50 µm

before annealing after annealing


-5.1

-5.0

-4.9

-4.8

Ther

moe

lect

ric P

ower

[µV

/°C

]

0

5

10

15

20

80 60 40 80 60 40Cold-rolling reduction [%]

Diff

eren

ce in

TEP

[%] gold tip reference

copper tip referencebefore annealing

after annealing

111

Noncontacting Thermoelectric Testerclosed-circuit Seebeck effect

• relative to surrounding intact material

• no artificial interface

• penetrating (with substantial depth)

• noncontact (with substantial lift-off)

specimen

heat

thermoelectric current

magnetometer

Material Effects versus GeometryTEP is independent of size and shape

C11000 copper

diameter 0.375”

∇T ≈ 0.5 °C/cm

2 mm lift-off distance

3” × 3” scanning dimension

18 nT peak magnetic flux

before annealing

after annealing

plastic zonemilled

T∇

pressed

T∇

112

Residual Stress Characterizationshot-peened C11000 copper

0

5

10

15

20

25

0 2A 4A 6A 8A 10A 12A 14A 16AAlmen Peening Intensity

Mag

netic

Sig

natu

re [n

T]before relaxationrelaxation at 235 ºCrelaxation at 275 ºCrelaxation at 315 °C2nd relaxation at 315 °C3rd relaxation at 460 °Crecrystallization at 600 °C

113

7 Electromagnetic Acoustic Transducers (EMATs)

7.1 EMAT Principles

7.2 EMAT Instrumentation

7.3 EMAT Applications

Piezoelectricity

+

-+

-

-

+

- -

+

+

- +-

- -

+

+

+

+ + + + + + +

- - - - - - -+ + + + + + +

- - - - - - -

SiSiSi

O OO

Quartz (silicon dioxide, SiO2)

E bV

Fσ = A

SE

eD ESe K

ε⎡ ⎤⎡ ⎤ ⎡ ⎤=σ ⎢− ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

114

Electromagnetic Acoustic Transducers

Key Features:

• non-contact/no couplant

• multiple wave modes (including SH)

Disadvantages:

• low sensitivity

• requires special electronics

• material dependent

Advantages:

• easy automation

• high speed scanning

• high reproducibility

• high-temperature inspection

• minimal wear

• less surface preparation required

• easy to customize

(EMATs)

7.1 EMAT Principles

115

Principle #1: Lorentz Force

( )Q= + ×F E v B

∇× =H J

t∂

∇× = −∂BE

= σJ E

Ampère's law:

Faraday's law:

Ohm’s law:

Lorentz force:

Je

conducting medium

Hp

He

Ip

Transmission (I ⇒ F) Reception (v ⇒ V):

Je

F

B0

I

B0

Je

v

V

Principle #2: MagnetizationFm magnetic force


V volume

M magnetization

H magnetic field

y height


specimen

electromagnet

y

200 2m

VdH dHF V Mdy dy

μ= −μ = −χ

excitation current magnetization force

no bias strong biassome bias

Time

Sign

al

Time

Sign

al

Time

Sign

al

Fm

116

Principle #3: Magnetostriction

Spontaneous magnetostriction:

H = 0

Induced magnetostriction:

H

1,2,3 3e

ε =

123e

ε = 12,3 2 3

eεε = − = −

0 2 4 6Magnetic Field [104 A/m]

low-carbon steel

Mag

neto

stric

tion

[10-6

]

-20

0

-10

-30

-40

10

Fe

Co

Ni

7.2 EMAT Instrumentation

117

EMAT Polarization

en dA Q≈ − = −I J vhigh coupling:

0n= − ×τ I B“surface” traction:

m 0Q= ×F v Bmagnetic force:

tangential polarization normal polarization

Je

B0 n I

τ τ

B0n I

Je

Normal-Beam EMATs

spiral coilradially polarized shear wave

rectangular coillinearly polarized shear wave

symmetric coillongitudinal wave

B0S

N

N SB0

SN

B0

118

Angle-Beam Shear EMATs

sin λθ =

Λ

periodic permanent magnethorizontally polarized shear wave

SN

Λ

θ

B0

meander coilvertically polarized shear wave

S

N

θ

Λ

B0

EMAT Electronics

EMATs withpermanent or

electromagnets

driveramplifieroscillator

+_

Vsmatchingnetwork

matchingnetwork

specimen

0 0.5 1 1.5 2 2.5 3Frequency [MHz]

02468

101214161820

Impe

danc

e [Ω

]

resistancereactance

7-turn, 10-mm-diameter spiral coil on ferritic steel

119

Impedance Matching

VgV

gZ

Z≈

( )2

g *max g g g

gwhen ,

8V

P Z Z R R X XR

= = = = −

transformer (κ ≈ 1)

12 2122 11

Φ Φ= = κ

Φ Φ

2 2 21 22( )dV Ndt

= Φ + Φ

1 1 11 12( )dV Ndt

= Φ + Φ

I1

N1 N2 V2

Φ11

V1

I2

Φ22Φ12 Φ21,

2 21 1

V NV N

=

2 2 11 22( )dV Ndt

= Φ + Φ

1 1 11 22( )dV Ndt

= Φ + Φ2 11 2

I NI N

=2

2 221 1

NZZ N

=

ideal transformer (κ = 1)

7.3 EMAT Applications

120

Texture Assessment by EMATscold-pressed 2024 aluminum, 1.4 MHz, EMAT

η ≈ 0% (annealed) η = 0.45 % η = 0.8 % η = 1.6 %

cavg = 2,850 m/s, 0.2% per division, η = (cmax – cmin)/cavg

Textured Specimen

transmitter receiver

Rayleighwave

High-Temperature Monitoring

60

55

50

45

40

35

30

25

20200 400 600 800 1000

Temperature [K]

Stiff

ness

[G

Pa]

C44

C66

230

210

190

170

150

130200 400 600 800 1000

Temperature [K]

Stiff

ness

[G

Pa]

C11

C33

SiC/Ti-6Al-4V composite

(Ogi et al., 2001)

121

Electromagnetic Acoustic Resonance(Hirao and Ogi et al., 2003)

0 50 100 150

SCM 440 steel pure titanium

-120 -80 -40 0Stress [MPa]

-1.0

-1.1

-1.2

Bire

fring

ence

[%

]

loadunload

couplantPZT

specimen specimen

EMAT

Stress [MPa]

0.05

0

-0.05

-0.10

Bire

fring

ence

[%

]

as-receivedquenched & temperedannealed

Thank You!

Documents

Electromagnetic NDE Lecture Slides